On the stability of free boundary minimal submanifolds in conformal domains

TL;DR

This paper proves that in certain conformal domains with convex boundaries, there are no stable free boundary minimal submanifolds of specific dimensions, under conditions of strict convexity or positive curvature.

Contribution

It establishes non-existence results for stable free boundary minimal submanifolds in conformal domains with convex boundaries, extending understanding of stability in geometric analysis.

Findings

No stable free boundary minimal submanifolds of dimension 2 to n-2 exist under given conditions.

Strict convexity or positive sectional curvature ensures stability restrictions.

Results apply to manifolds conformal to Euclidean convex domains.

Abstract

Given a $n$-dimensional Riemannian manifold with non-negative sectional curvatures and convex boundary, that is conformal to an Euclidean convex bounded domain, we show that it does not contain any compact stable free boundary minimal submanifold of dimension $2\leq k\leq n-2$, provided that either the boundary is strictly convex with respect to any of the two metrics or the sectional curvatures are strictly positive.